Copyright 1996 Lawrence C. Marsh 0 PowerPoint Slides for Undergraduate Econometrics by

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allChap6

i

)

f(x

i

) +

Σ

g

2

(x

i

)

f(x

i

)

n

i = 1

n

i = 1

E

[g(X)]

=

E

1

(X)]

+ E

2

(X)]

2.23

Copyright 1996 Lawrence C. Marsh

Adding

and

Subtracting

Random Variables

E(X-Y) = E(X) - E(Y)

E(X+Y) = E(X) + E(Y)

2.24

Copyright 1996 Lawrence C. Marsh

E(X+a) = E(X) + a

Adding

constant

to a variable will

add a constant to its expected value:

Multiplying by

constant

will multiply

its expected value by that constant:

E(bX) = b E(X)

2.25

var(X)

= average squared deviations

around the mean of X.

var(X)

= expected value of the squared deviations

around the expected value of X.

var(X) =

E [(X - EX) ]

Variance

2.26

8

Copyright 1996 Lawrence C. Marsh

var(X) =

E [(X - EX) ]

E [X - 2XEX + (EX) ]

E(X ) - 2 EX EX + E (EX)

E(X ) - 2 (EX) + (EX)

2

2

E(X ) - (EX)

var(X) =

E [(X - EX) ]

var(X) = E(X ) - (EX)

2.27

variance

of a discrete

random variable, X:

standard deviation is square root of variance

var ( X ) =

i

- EX )

f(x

)

i = 1

∑

2.28

f(x

) (x

- EX) (x

- EX) f(x

)

2 - 4.3 = -2.3

5.29 (.1) = .529

3 - 4.3 = -1.3

1.69 (.3) = .507

.1

4 - 4.3 = - .3

.09 (.1) = .009

5 - 4.3 = .7

.49 (.2) = .098

.3

6 - 4.3 = 1.7

2.89 (.3) = .867

f(x

) = .2 + .9 + .4 + 1.0 + 1.8 = 4.3

Σ

(x

- EX) f(x

) = .529 + .507 + .009 + .098 + .867

= 2.01

2

calculate the

variance

for a

discrete random variable, X:

i = 1

n

n

i = 1

2.29

Z = a + cX

var(Z) = var(a + cX)

= E [(a+cX) - E(a+cX)]

= c var(X)

var(a + cX) = c var(X)

2.30

joint

probability density function,

f(x,y), provides the probabilities

associated with the joint occurrence

of all of the possible pairs of X and Y.

Joint pdf

2.31

Copyright 1996 Lawrence C. Marsh

college grads

in household

.15

.05

.45

.35

joint pdf

f(x,y)

Y = 1

Y = 2

vacation

homes

owned

X = 0

X = 1

Survey of College City, NY

(0,1)

(0,2)

(1,1)

(1,2)

2.32

9

Copyright 1996 Lawrence C. Marsh

E[g(X,Y)] =

g(x

) f(x

)

E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=

.75

E(XY) =

f(x

)

Calculating the expected value of

functions of two random variables.

2.33

The

marginal

probability density functions,

f(x) and f(y), for discrete random variables,

can be obtained by summing over the f(x,y)

with respect to the values of Y to obtain f(x)

with respect to the values of X to obtain f(y).

f(x

i

) =

f(x

)

f(y

) =

f(x

)

Marginal pdf

2.34

.15

.05

.45

.35

marginal

Y = 1

Y = 2

X = 0

X = 1

.60

.40

.50

(X = 1)

(X = 0)

(Y = 1)

(Y = 2)

marginal

pdf for Y:

marginal

pdf for X:

2.35

Copyright 1996 Lawrence C. Marsh

The

conditional

probability density

functions of X given Y=y , f(x

y),

and of Y given X=x , f(y

x),

are obtained by dividing f(x,y) by f(y)

to get f(x

y) and by f(x) to get f(y

x).

f(x

y) =

f(y

x) =

f(x,y)

f(y)

f(x)

Conditional pdf

2.36

.15

.05

.45

.35

conditonal

Y = 1

Y = 2

X = 0

X = 1

.60

.40

.50

.25

.75

.875

.125

.90

.10

.70

.30

(Y=2

X= 0)=.25

(Y=1

X = 0)=.75

(Y=2

X = 1)=.875

(X=0

Y=2)=.30

(X=1

Y=2)=.70

(X=0

Y=1)=.90

(X=1

Y=1)=.10

(Y=1

X = 1)=.125

2.37

Copyright 1996 Lawrence C. Marsh

X and Y are

independent

random

variables if their joint pdf, f(x,y),

is the product of their respective

marginal pdfs, f(x) and f(y) .

f(x

) = f(x

) f(y

)

for

independence

this must hold for all pairs of i and j

Independence

2.38

10

Copyright 1996 Lawrence C. Marsh

.15

.05

.45

.35

not independent

Y = 1

Y = 2

X = 0

X = 1

.60

.40

.50

(X = 1)

(X = 0)

(Y = 1)

(Y = 2)

marginal

pdf for Y:

marginal

pdf for X:

.50x.60=

.30

.50x.60=

.30

.50x.40=

.20

.50x.40=

.20

The calculations

in the boxes show

the numbers

required to have

independence

2.39

The

covariance

between two random

variables, X and Y, measures the

linear association between them.

cov(X,Y) = E[(X - EX)(Y-EY)]

Note that variance is a special case of covariance.

cov(X,X) = var(X) = E[(X - EX) ]

2

Covariance

2.40

Copyright 1996 Lawrence C. Marsh

cov(X,Y) =

E [(X - EX)(Y-EY)]

E [XY - X EY - Y EX + EX EY]

E(XY) - 2 EX EY + EX EY

E(XY) - EX EY

cov(X,Y) =

E [(X - EX)(Y-EY)]

cov(X,Y) = E(XY) - EX EY

E(XY) - EX EY - EY EX + EX EY

2.41

.15

.05

.45

.35

Y = 1

Y = 2

X = 0

X = 1

.60

.40

.50

EX=0(.60)+1(.40)=

.40

EY=1(.50)+2(.50)=

1.50

E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=

.75

EX EY = (.40)(1.50) =

.60

cov(X,Y) = E(XY)

- EX EY

= .75 - (.40)(1.50)

= .75 - .60

.15

covariance

2.42

Copyright 1996 Lawrence C. Marsh

The

correlation

between two random

variables X and Y is their covariance

divided by the square roots of their

respective variances.

Correlation

is a pure number falling between -1 and 1.

cov(X,Y)

(X,Y) =

var(X) var(Y)

Correlation

2.43

Copyright 1996 Lawrence C. Marsh

.15

.05

.45

.35

Y = 1

Y = 2

X = 0

X = 1

.60

.40

.50

EX=

.40

EY=

1.50

cov(X,Y) =

.15

correlation

EX=0(.60)+1(.40)=

.40

var(X) = E(X )

- (EX)

= .40 - (.40)

.24

2

2

EY=1(.50)+2(.50)

= .50 + 2.0

2.50

2

2

var(Y) = E(Y )

- (EY)

= 2.50 - (1.50)

.25

(X,Y) =

cov(X,Y)

var(X) var(Y)

(X,Y) =

.61

2.44

11

Copyright 1996 Lawrence C. Marsh

Independent random variables

have zero covariance and,

therefore, zero correlation.

The converse is not true.

Zero Covariance & Correlation

2.45

Copyright 1996 Lawrence C. Marsh

The expected value of the weighted sum

of random variables is the sum of the

expectations of the individual terms.

Since expectation is a linear operator,

it can be applied term by term.

E[c

1

X + c

Y] = c

EX + c

E[c

+...+ c

] = c

+...+ c

In general, for random variables X

, . . . , X

2.46

The

variance of a weighted sum

of random

variables is the sum of the variances, each times

the square of the weight, plus twice the covariances

of all the random variables times the products of

their weights.

var(c

1

X

Y)=c

var(X)+c

var(Y)

cov(X,Y)

var(c

−

Y) = c

var(X)+c

var(Y)

−

cov(X,Y)

Weighted

sum

of random variables:

Weighted

difference

of random variables:

2.47

Copyright 1996 Lawrence C. Marsh

The Normal Distribution

Y ~ N(

β

,

)

f(y) =

exp

f(y)

(y -

)

2.48

The Standardized Normal

Z ~ N(

0

,

)

f(z) =

exp

Z = (y -

)/

σ

2.49

Copyright 1996 Lawrence C. Marsh

P

[

Y > a

]

= P > = P Z >

a -

β

a -

Y -

f(y)

Y ~ N(

)

2.50

12

Copyright 1996 Lawrence C. Marsh

P

[

a < Y < b

]

= P < <

= P < Z <

a -

β

Y -

b -

a -

b -

f(y)

Y ~ N(

)

2.51

Copyright 1996 Lawrence C. Marsh

~ N(

), Y

~ N(

), . . . , Y

~ N(

)

W = c

+ c

+ . . . + c

Y

n

Linear combinations of jointly

normally distributed random variables

are themselves normally distributed.

W ~ N

[

E(W), var(W)

]

2.52

mean: E[V] = E[

(m)

] = m

If Z

, Z

, . . . , Z

denote m independent

N(0,1) random variables, and

V = Z

+ Z

+ . . . + Z

, then V ~

(m)

2

2

2

2

V is

chi-square

with m degrees of freedom.

Chi-Square

variance: var[V] = var[

(m)

] = 2m

If Z

, Z

, . . . , Z

denote m independent

N(0,1) random variables, and

V = Z

+ Z

+ . . . + Z

, then V ~

(m)

2

2

2

2

V is

chi-square

with m degrees of freedom.

2

2

2.53

mean: E[

] = E[

(m)

] = 0 symmetric about zero

variance: var[

] = var[

(m)

] = m / (m

−

2

)

If Z ~ N(0,1) and V ~

(m)

and if Z and V

are independent then,

(m)

student-t

with m degrees of freedom.

t =

Student - t

2.54

If V

(m

1

)

and V

(m

2

)

and if V

and V

are independent, then

~ F

(m

1

,m

2

)

is an

F statistic

with m

numerator

degrees of freedom and m

denominator

degrees of freedom.

2

F =

1 m

m

2

2

F Statistic

2.55

Copyright 1996 Lawrence C. Marsh

The Simple Linear

Regression

Model

Chapter 3

that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the

John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution

or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these

programs or from the use of the information contained herein.

3.1

13

Copyright 1996 Lawrence C. Marsh

1. Estimate

a relationship

among economic

variables, such as y = f(x).

2. Forecast or

predict

the value of one

variable, y, based on the value of

another variable, x.

Purpose of Regression Analysis

3.2

y = dollars spent each week on food items.

x = consumer’s weekly income.

The relationship between x and the expected

value of y , given x, might be

linear

E(y|x) =

3.3

f(y|x=480)

y|x=480

Figure 3.1a Probability Distribution f(y|x=480)

of Food Expenditures if given income x=$480.

3.4

f(y|x)

f(y|x=480)

f(y|x=800)

µ

y|x=480

y|x=800

Figure 3.1b Probability Distribution of Food

Expenditures if given income x=$480 and x=$800.

3.5

Copyright 1996 Lawrence C. Marsh

{

∆

E(y|x)

Average

Expenditure

x (income)

E(y|x)=

β

1

∆

E(y|x)

∆

Figure 3.2 The Economic Model: a linear relationship

between avearage expenditure on food and income.

3.6

=480

=800

f(y

)

Figure 3.3. The probability density function

for y

at two levels of household income, x

expenditure

Homoskedastic Case

income

3.7

14

Copyright 1996 Lawrence C. Marsh

.

f(y

)

Figure 3.3+. The variance of y

increases

as household income, x

, increases.

expenditure

Heteroskedastic Case

.

.

income

3.8

Copyright 1996 Lawrence C. Marsh

Assumptions of the Simple Linear

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